**Karl Weierstrass** is best known for his construction of the theory of complex functions by means of power series.

- Wilhelm Weierstrass was a well educated man who had a broad knowledge of the arts and of the sciences.
- He certainly was well capable of attaining higher positions than he did, and this attitude may have been one of the reasons that Karl Weierstrass's early career was in posts well below his outstanding ability.
- Wilhelm Weierstrass became a tax inspector when Karl was eight years old.
- By 1829 Wilhelm Weierstrass had become an assistant at the main tax office in Paderborn, and Karl entered the Catholic Gymnasium there.
- While at the Gymnasium Weierstrass certainly reached a level of mathematical competence far beyond what would have been expected.
- The result of the conflict which went on inside Weierstrass was that he did not attend either the mathematics lectures or the lectures of his planned course.
- Weierstrass had made a decision to become a mathematician but he was still supposed to be on a course studying public finance and administration.
- On 22 May 1839 Weierstrass enrolled at the Academy in Münster.
- Gudermann lectured in Münster and this was the reason that Weierstrass was so keen to study there.
- Weierstrass attended Gudermann's lectures on elliptic functions, some of the first lectures on this topic to be given, and Gudermann strongly encouraged Weierstrass in his mathematical studies.
- Leaving Münster in the autumn of 1839, Weierstrass studied for the teacher's examination which he registered for in March 1840.
- At Weierstrass's request he was given a question on the paper he received in May 1840 on the representation of elliptic functions and he presented his own important research as an answer.
- When, in later life, Weierstrass learnt of Gudermann's comments he said that he would have published his results had he known.
- Weierstrass also commented on how generous Gudermann had been in his praise, particularly since he had been highly critical of Gudermann's methods.
- By April 1841 Weierstrass had taken the necessary oral examinations and he began one year probation as a teacher at the Gymnasium in Münster.
- The transformation of his conception of an analytic function from a differentiable function to a function expansible into a convergent power series was made during this early period of Weierstrass's mathematical activity.
- Weierstrass began his career as a qualified teacher of mathematics at the Pro-Gymnasium in Deutsch Krone in West Prussia (now Poland) in 1842 where he remained until he moved to the Collegium Hoseanum in Braunsberg in 1848.
- As a teacher of mathematics he was required to teach other topics too, and Weierstrass taught physics, botany, geography, history, German, calligraphy and even gymnastics.
- From around 1850 Weierstrass began to suffer from attacks of dizziness which were very severe and which ended after about an hour in violent sickness.
- It is not surprising that when Weierstrass published papers on abelian functions in the Braunsberg school prospectus they went unnoticed by mathematicians.
- This paper did not give the full theory of inversion of hyperelliptic integrals that Weierstrass had developed but rather gave a preliminary description of his methods involving representing abelian functions as constantly converging power series.
- With this paper Weierstrass burst from obscurity.
- In 1855 Weierstrass applied for the chair at the University of Breslau left vacant when Kummer moved to Berlin.
- Kummer, however, tried to influence things so that Weierstrass would go to Berlin, not Breslau, so Weierstrass was not appointed.
- A letter from Dirichlet to the Prussian Minister of Culture written in 1855 strongly supported Weierstrass being given a university appointment.
- After being promoted to senior lecturer at Braunsberg, Weierstrass obtained a year's leave of absence to devote himself to advanced mathematical study.
- Weierstrass published a full version of his theory of inversion of hyperelliptic integrals in his next paper Theorie der Abelschen Functionen Ⓣ(Theory of abelian functions) in Crelle's Journal in 1856.
- Although he would have prefered to go to the University of Berlin, Weierstrass certainly did not want to return to the Collegium Hoseanum in Braunsberg so he accepted the offer from the Institute on 14 June 1856.
- Offers continued to be made to Weierstrass so that when he attended a conference in Vienna in September 1856 he was offered a chair at any Austrian university of his choice.
- Weierstrass's successful lectures in mathematics attracted students from all over the world.
- In his lectures of 1859/60 Weierstrass gave Introduction to analysis where he tackled the foundations of the subject for the first time.
- We described above the health problems that Weierstrass suffered from 1850 onwards.
- In his 1863/64 course on The general theory of analytic functions Weierstrass began to formulate his theory of the real numbers.
- Weierstrass's lectures developed into a four-semester course which he continued to give until 1890.
- Weierstrass's approach still dominates teaching analysis today and this is clearly seen from the contents and style of these lectures, particularly the Introduction course.
- Its contents were: numbers, the function concept with Weierstrass's power series approach, continuity and differentiability, analytic continuation, points of singularity, analytic functions of several variables, in particular Weierstrass's "preparation theorem", and contour integrals.
- At Berlin, Weierstrass had two colleagues Kummer and Kronecker and together the three gave Berlin a reputation as the leading university at which to study mathematics.
- Kronecker was a close friend of Weierstrass's for many years but in 1877 Kronecker's opposition to Cantor's work cause a rift between the two men.
- This became so bad that at one stage, in 1885, Weierstrass decided to leave Berlin and go to Switzerland.
- A large number of students benefited from Weierstrass's teaching.
- In 1870 Sofia Kovalevskaya came to Berlin and Weierstrass taught her privately since she was not allowed admission to the university.
- It was through Weierstrass's efforts that Kovalevskaya received an honorary doctorate from Göttingen, and he also used his influence to help her obtain the post in Stockholm in 1883.
- Weierstrass and Kovalevskaya corresponded for 20 years between 1871 to 1890.
- He decided to supervise the publication of his own complete works, in his case this would involve a great deal of unpublished material from his lecture courses and Weierstrass realised that without his help this would be a difficult task.

Born 31 October 1815, Ostenfelde, Westphalia (now Germany). Died 19 February 1897, Berlin, Germany.

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Algebra, Analysis, Geometry, Origin Germany, Set Theory, Topology

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive